Ambarzumyan's theorem with eigenparameter in the boundary conditions

Abstract

In this paper, the classical Ambarzumyan's theorem for the regular Sturm-Liouville problem is extended to the case in which the boundary conditions are eigenparameter dependent. Specifically, we show that if the spectrum of the operator − D 2 + q with eigenparameter dependent boundary conditions is the same as the spectrum belonging to the zero potential, then the potential function q is actually zero.